## Journal of Applied Mathematics

### Network flow optimization for restoration of images

Boris A. Zalesky

#### Abstract

The network flow optimization approach is offered for restoration of gray-scale and color images corrupted by noise. The Ising models are used as a statistical background of the proposed method. We present the new multiresolution network flow minimum cut algorithm, which is especially efficient in identification of the maximum a posteriori (MAP) estimates of corrupted images. The algorithm is able to compute the MAP estimates of large-size images and can be used in a concurrent mode. We also consider the problem of integer minimization of two functions, $U_1(\mathbf{x}) = \lambda\sum_i|y_i-x_i| +\sum_{i,j}\beta_{i,j}|x_i-x_j|$ and $U_2(\mathbf{x}) =\sum_i\lambda_i(y_i-x_i)^2 +\sum_{i,j}\beta_{i,j}(x_i-x_j)^2$, with parameters $\lambda,\lambda_i,\beta_{i,j} > 0$ and vectors $\mathbf{x}=(x_1,\dotsc,x_n)$, $\mathbf{y} = (y_1,\dotsc,y_n)\in\{0,\dotsc,L-1\}^n$. Those functions constitute the energy ones for the Ising model of color and gray-scale images. In the case $L = 2$, they coincide, determining the energy function of the Ising model of binary images, and their minimization becomes equivalent to the network flow minimum cut problem. The efficient integer minimization of $U_1(\mathbf{x}),U_2(\mathbf{x})$ by the network flow algorithms is described.

#### Article information

Source
J. Appl. Math., Volume 2, Number 4 (2002), 199-218.

Dates
First available in Project Euclid: 30 March 2003

https://projecteuclid.org/euclid.jam/1049074994

Digital Object Identifier
doi:10.1155/S1110757X02110035

Mathematical Reviews number (MathSciNet)
MR1948085

Zentralblatt MATH identifier
1116.90321

#### Citation

Zalesky, Boris A. Network flow optimization for restoration of images. J. Appl. Math. 2 (2002), no. 4, 199--218. doi:10.1155/S1110757X02110035. https://projecteuclid.org/euclid.jam/1049074994

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